4, A given data questions. a) What are the new values of the mean and standard d

Question

4, A given data questions. a) What are the new values of the mean and standard deviation if the same constant k is added to set has a mean and standard deviation . Answer each of the following each data value in the given set? Explain. (6 pts.) b) What are the new values of the mean and standard deviation if each data value of the data set is multiplied by the same constant k? Explain. (6 pts.) c) If the standard deviation of a given data set is zero, what can you say about the data values in the data set? (4 pts.) jt mee^ sally4hesw v/Jbe7er,aral/

Explanation / Answer

We limit the discussion to a data set with 3 values for simplicity, but the conclusions are true for any data set with quantitative data. Let x, y and z be the data values making a data set. The mean = (x + y + z) / 3 The standard deviation = [ ((x - )2 + (y - )2 + (z - )2)/3 ] We now add a constant k to each data value and calculate the new mean '. ' = ((x + k) + (y + k) + (z + k)) / 3 = (x + y + z) / 3 + 3k/3 = + k We now calculate the new mean standard deviation '. ' = [ ((x + k - ')2 +(y + k - ')2+(z + k - ')2)/3 ] Note that x + k - ' = x + k - - k = x - also y + k - ' = y + k - - k = y - and z + k - ' = z + k - - k = z - Therefore ' = [ ((x - )2 +(y - )2+(z - )2)/3 ] = If we add the same constant k to all data values included in a data set, we obtain a new data set whose mean is the mean of the original data set PLUS k. The standard deviation does not change. We now multiply all data values by a constant k and calculate the new mean ' and the new standard deviation '. ' = (kx + ky + kz) / 3 = k ' = [ ((kx - k)2 +(ky - k)2+(kz - k)2)/3 ] = |k| If we multiply all data values included in a data set by a constant k, we obtain a new data set whose mean is the mean of the original data set TIMES k and standard deviation is the standard deviation of the original data set TIMES the absolute value of k.

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